Representations and properties of a new family of ω-Caputo fractional derivatives

Abstract

In the most general case of ω-weights, some normed functional spaces $X_omega^p$(a, b)(1 ≤ p ≤ ∞), $AC_{gamma,omega}^eta$[a, b] anda generalization of the fractional integro-differentiation operator are introduced and analyzed. The boundedness of theω-weighted fractional operator over $X_omega^p$(a, b) is proved. Some theorems and lemmas on the properties of the invertions ofthe mentioned operator and several representations of functions from $AC_{gamma,omega}^eta$[a, b] are established. A general ω-weightedCaputo fractional derivative of order α is studied over $AC_{gamma,omega}^eta$[a, b]. Some representations and other properties of thisfractional derivative are proved. Some conclusions are presented.